A solution to this problem was found by
Gálvez, Kock, and Tonks [1]}, [2]}, [3]}. They discovered
that the incidence coalgebra construction and Möbius inversion
make sense for objects more general than Möbius categories (recall
that Möbius categories include locally finite posets and
Cartier–Foata monoids). These are completely new objects in this context which they
call decomposition spaces. They are certain simplicial objects
subject to an axiom that expresses decomposition, in the same way as
the Segal condition (which characterises categories among simplicial
sets) expresses composition. Decomposition spaces are the same thing
as the 2-Segal spaces of Dyckerhoff and Kapranov [4]} (see
[5]} for the last piece of this equivalence). It seems
likely that all combinatorial coalgebras, bialgebras and Hopf
algebras (with nonnegative section coefficients) arise from the
incidence coalgebra construction of decomposition spaces. This has
been shown for most of Schmitt's examples [6]} (restriction species in
Gálvez–Kock–Tonks [7]} and hereditary species in
Carlier [8]}). Gálvez–Kock–Tonks (as also
Dyckerhoff–Kapranov) work in the fully homotopical setting of
simplicial \(\infty \) -groupoids, but already the discrete case of the
notion is very rich, as exemplified by work of Bergner et
al. [9]} and Kock–Spivak [10]}, who relate
the notion to constructions in algebraic topology and category
theory.
